DLMF:18.26.E18 (Q5948)

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DLMF:18.26.E18
No description defined

    Statements

    Defining formula
    F 1 2 ⁡ ( a + i ⁢ y , d + i ⁢ y a + d ; z ) ⁢ F 1 2 ⁡ ( b - i ⁢ y , c - i ⁢ y b + c ; z ) = ∑ n = 0 ∞ W n ⁡ ( y 2 ; a , b , c , d ) ( a + d ) n ⁢ ( b + c ) n ⁢ n ! ⁢ z n , Gauss-hypergeometric-F-as-2F1 𝑎 𝑖 𝑦 𝑑 𝑖 𝑦 𝑎 𝑑 𝑧 Gauss-hypergeometric-F-as-2F1 𝑏 𝑖 𝑦 𝑐 𝑖 𝑦 𝑏 𝑐 𝑧 superscript subscript 𝑛 0 Wilson-polynomial-W 𝑛 superscript 𝑦 2 𝑎 𝑏 𝑐 𝑑 Pochhammer 𝑎 𝑑 𝑛 Pochhammer 𝑏 𝑐 𝑛 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left({a+iy,d+iy\atop a+d};z\right){{% }_{2}F_{1}}\left({b-iy,c-iy\atop b+c};z\right)=\sum_{n=0}^{\infty}\frac{W_{n}% \left(y^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}{\left(b+c\right)_{n}}n!}z^{% n},}}
    0 references
    DLMF id
    DLMF:18.26.E18
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    Q11783
    Defining formula
    W n ⁡ ( x ; a , b , c , d ) Wilson-polynomial-W 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c},\NVar{d}\right)}}
    xml-id
    C18.S25.T1.t1.r2.m2aedec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1hdec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1dec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1pdec
    0 references
     
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